MATLAB: 2nd order ode by ode45 or other numerical method

2nd order ode

Hi every one,
I am trying to solve below differential equation with matlab:
y''+(2/x)*y'=0.1248y
B.C 1 : y'(0)=0
B.C 2: y(1)=1
x interval: [0 1]
solution interval (or y): [0 1]
I used bvp4c and ode45 but every time I faced with error like "solve the collocation equations — a singular Jacobian encountered"
this is my code:
function SolveBVP()
solinit = bvpinit([0,1],[0 0]);
sol = bvp4c(@deriv,@bcs,solinit);
x=linspace(0,1,100);
y=deval(sol,x);
plot(x,y(1,:),'b-x');
 
function dYdx = deriv(x,Y)
 
dYdx(1) = Y(2); 
dYdx(2) = 0.1248*Y(1)-(2/x)*Y(2);
function res = bcs(ya,yb)
res = [ ya(2) 
        yb(1)-1];
It would be appricated if any one of you can help me.
best regards

Best Answer

You ODE have term 2/x, which becomes infinite at x=0. To avoid this, start the limit a bit above 0 (say 1e-20). Also, it is better to pass the initial guess at several points. Try this
x=linspace(1e-20,1,100);
solinit = bvpinit(x,[1 0]);
sol = bvp4c(@deriv,@bcs,solinit);
y=deval(sol,x);
plot(x,y(1,:),'b-x');
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